Aircraft are divided into two areas for licensing and airworthiness purposes:

EASA aircraft

non-EASA aircraft

This classification applies to types of aircraft, not individual aircraft. So, for example, if a particular Cessna 172N is an EASA aircraft then all Cessna 172N are EASA aircraft. And if one particular De Havilland Chipmunk T10 is not an EASA aircraft all such Chipmunks are classed as being non-EASA. Non-EASA aircraft are also known as Annex II (two) aircraft.

EASA aircraft

Many aircraft in Europe are classed as EASA aircraft wherever they may have been manufactured or registered. This includes many of the types you'll see around flying schools – like the Cessna range, the Piper PA-28s and PA-38s, Cirrus etc.
In the UK, holders of Part-FCL EASA licences can fly both EASA and UK-registered non-EASA aircraft that are within the ratings included in their licence.

For example: The Cessna 172 is an EASA aircraft. The Tiger Moth is a non-EASA aircraft. Both are single engine piston aircraft. So if you have a Part-FCL licence, like a PPL(A) or LAPL(A) that allows you to fly with a single-engine piston rating you can fly both the Cessna 172 (EASA) and the Tiger Moth (non-EASA). But if you have a national licence, such as the UK NPPL(SSEA), after April 8th 2015 you can only fly the Tiger Moth.

Non-EASA aircraft

With some exceptions, the following types of aircraft are defined as non-EASA aircraft and are ruled by national, not European, regulations:

Microlights

Light gyroplanes

Ex-military aircraft

Foot-launched aircraft

Vintage aircraft

You do not have to have an EASA licence to fly these types of aircraft as you can fly them if you only have a national licence.